How do you make an animated mist / smoke with lights that go through it and interact with it?
Final result:

First thoughts:
This is pretty complex setup and very demanding for the hardware. It's made using Cycles. You should have some knowledge in such fields as:

creating smoke,
setting up materials,
render layers,
basic compositing.

Final result is made with low settings.
Scene setup:
First of all we need to setup up our scene.

Crank up Volume Light Paths. I've used 4. Higher values gives you better light scattering trough smoke. This step is essential for the whole thing.



Make Environment color black, or whatever you need in your case.

Object setup:
In my scene I've used two Ngon filled Circles, one for the light, and second for the Smoke Flow. One Empty for Force Field. Cube for Smoke Domain.

Light:
Small Circle behind Smoke Domain with Emission shader. It could be Sphere or anything else.
This is node setup I've used for it:

Smoke Flow:
Again it's Circle, and again you can use different object. If you for example want to have really dense smoke, you can use Sphere for it.
Note 1: Remember to turn off view and render of this mesh in Outliner or assign to it Transparent Shader!
Set Temp Diff. really low to keep smoke in place. In this case it's 0.01.
Note 2: Denser smoke will need higher Volume Light Paths and/or smaller smoke material density (we will get to it).

Smoke Domain:
This is two step process.
1. Smoke Domain setup:
Nothing special here. Only Noise Method is worth mentioning, lower setting will give you smoother smoke.

2. Smoke Domain material:
This is almost default material after using Quick Smoke, with addition of Brightness/Contrast node, which gives us harder and wither feel. Also default density value is decreased (this is connected to Smoke Flow Note 1 from above).

Force Field:
To make some smoke movement I've used Force Field &gt; Turbulence. I like to stop smoke flow then move it by a force in such cases. Force field is assigned to an Empty.

Render Layers and Composite:
To make some light blur and streaks you will need to move light emitter to another layer.
In this case I've moved my light mesh to second layer and kept other object on first layer then made two Render Layers. First Smoke, which is rendering first layer, and second - Light which is rendering second layer.
 
After final rendering, go to Composite nodes and make this setup. This is just blurred out light with streaks and gamma correction of smoke and light.

Final thoughts:
As I said at the beginning, this isn't easy if you're new to Blender. You can find any information related to techniques used here or you can just ask me and I will extend this as needed.
You need to have in mind that values used in this particular setup will vary in any other. Things like density, emission strength, smoke resolution, volume light paths etc. could be much different then these.
Check my blend file. Load it, check it, quick rewrite it.
Blend file:

Edit:
You need to use for this filmic-blender. You should always use it.